Systems and methods for determining intraocular lens power

ABSTRACT

A system for providing an intraocular lens includes a processor and a computer readable memory. The computer readable memory is configured to communicate with the processor, the memory having stored therein at least one of: one or more ocular dimensions, and at least one predetermined refractive outcome. The memory further includes a sequence of instructions which, when executed by the processor, cause the processor to select an intraocular lens, select a power of an intraocular lens, or provide an intraocular lens. The sequence of instructions includes determining one or more dimensions of an eye. The instructions also include calculating, based on a mathematical relationship, a distance from an apex of a cornea of the eye to an apex or plane of the intraocular lens after insertion into the eye. The instructions further calculating an optical power of the intraocular lens suitable for providing a predetermined refractive outcome. The mathematical relationship includes an axial length of the eye, an anterior chamber depth of the natural crystalline lens, and a corneal radius of the eye, but is independent of a thickness of the natural crystalline lens.

RELATED APPLICATION

The present application claims priority under 35 U.S.C §119(e) toprovisional application No. 61/042,697, filed on Apr. 4, 2008, theentire contents of which applications is hereby incorporated byreference in its entirety for all purposes as if fully set forth herein.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates generally to ocular surgical proceduresinvolving implantable or injectable lenses into the eye of a recipient,and more specifically to systems and methods for determination orselection of a lens power for providing emmetropic vision or, if chosen,a specific ametropic vision to a pseudophakic eye.

2. Description of the Related Art

Accurate determination of lens power is an important aspect in providingemmetropia, or a desired degree of ametropia, to a subject undergoingcataract surgery or other ophthalmic procedures in which the naturalcrystalline lens is replaced with or supplemented by implantation of anintraocular lens (IOL) into the eye. Measurements of the eye aretypically made preoperatively and a lens power is selected based oncorrelations between the measured values and lens powers providing adesired refractive outcome.

Over the years a number of intraocular lens power calculation formulashave been developed, for example, as discussed in the book published bySLACK Incorporated entitled Intraocular Lens Power Calculations, by H.John Shammas, which is herein incorporated by reference in its entirety.These power formulas may be broadly characterized into at least twocategories: theoretical formulas, which are based on a geometric optic,two-lens vergence formula; and regression formulas, which are based onregression formulas obtained by fitting data from a large patientdatabase to an equation relating lens power to one or more parametersthought to correlate with lens power. While there has been continuedprogress in the accuracy of intraocular lens power calculation formulasto obtain better refractive outcomes, undesirable refractive outcomesdue to improper intraocular lens power calculations are still relativelycommon. Apart from the general desire for spectacle-free refractiveoutcomes, demands for more accurate lens power calculation have alsoincreased due to the introduction of multifocal lenses.

Many of the current formula algorithms were derived by opticalback-calculation to agree with the refractive outcome. In this mannerthey may be confounded with errors in all parameters used in thecalculation, and the oversimplification of thin-lens theory. Anevaluation of the sources of errors in lens power calculations wasrecently published by one of the current co-inventors (Sverker Norrby,“Sources of error in intraocular lens power calculation”, Journal ofCataract and Refractive Surgery, Vol. 34, pp. 368-376, March 2008, whichis herein incorporated by reference in its entirety). In this paper,preoperative estimation of postoperative intraocular lens position wasdetermined to be the largest contributor of error in the refractiveoutcome of cataract surgery, with an error contribution of 35%, relativeto all error sources evaluated.

In most, if not all of the current formula algorithms, there are anumber of ocular parameters that are used in deriving an appropriatelens power for implantation into the eye. These parameters include axiallength (AL), corneal radius (CR) or power (K), and anterior chamberdepth of the natural crystalline lens prior to surgery (ACD_(pre)). Ingeneral, one or more of these parameters are used to provide thepreoperative estimation of postoperative intraocular lens position. Theestimated postoperative lens position is then used in combination withone or more of these same parameters to provide an estimate of thecorrect lens power to provide a desired refractive outcome (typicallyemmetropia). However, as discussed in the previous paragraph, the use ofthis term in calculating postoperative lens position is a large errorsource in this process. In addition, some of these parameters may beunavailable at the time of evaluation. For example, in the case of apatient that has previously received a corneal refractive surgery, suchas LASIK or PRK, the original corneal radius or power may no longer beavailable. It is the corneal radius prior to corneal refractive surgerythat is correct to use in the power calculation formulas because theywere developed for normal eyes on data pertaining to normal eyes. Thecorneal refractive surgery has changed the anatomic relations of oculardimensions. Hence, the CR or K determined for corneas that have hadcorneal refractive surgery will give erroneous estimates of the IOLposition.

Accordingly, better systems and methods are needed that will allowreliable and accurate determination of implanted or injected lens power,and to provide such determination even in cases where knowledge ofinformation such as original corneal radius or power is no longeravailable.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments of the present invention may be better understood from thefollowing detailed description when read in conjunction with theaccompanying drawings. Such embodiments, which are for illustrativepurposes only, depict novel and non-obvious aspects of the invention.The drawings include the following figures:

FIG. 1 is a cross-sectional view of a phakic eye containing a naturalcrystalline lens.

FIG. 2 is a cross-sectional view of a pseudophakic eye containing anintraocular lens.

FIG. 3 is a flow chart of a method according to an embodiment of thepresent invention.

FIG. 4 is a graphical representation of the elements of computing systemfor selecting an ophthalmic lens according to an embodiment of thepresent invention.

DETAILED DESCRIPTION OF THE DRAWINGS

The present invention generally provides devices, systems, and methodsfor selecting ophthalmic lenses and/or an optical power for such lensesthat will provide a predetermined refractive outcome. In many cases thedesired outcome will be emmetropia, for example, so that the eye intowhich the lens is located has a visual acuity for distant objects thatis at least 20/20 based on a Snellen chart and/or has an ModulationTransfer Function (MTF) that is at least 0.15 or at least 0.20 at aspatial frequency of 50 line pairs per mm. The following disclosure willbe primarily directed to embodiments of the invention as they apply toimplantable intraocular lenses; however, it will be understood thatother embodiments may be applied directly, or indirectly, to other typesof ophthalmic lenses including, but not limited to, injectable lensesthat may be formed by injection of a polymer fluid or gel into the eyewhich is subsequently cured or solidified (e.g., using a UV source),corneal implants, corneal surgical procedures such as LASIK or PRK,contact lenses, and other such devices. In some embodiments, varioustypes of ophthalmic devices are combined, for example, an intraocularlens and a LASIK procedure may be used together to provide apredetermined visual outcome. Embodiments of the invention may findparticular use with multifocal or accommodating intraocular lenses,where a proper selection of lens power may be particular important forachieving a desired refractive outcome.

Referring to FIG. 1, which is a cross-sectional view of a phakic eyewith the natural crystalline lens, an eye 10 comprises a retina 12 thatreceives light in the form of an image that is produced by thecombination of the optical powers of a cornea 14 and a naturalcrystalline lens 16, both of which are generally disposed about anoptical axis OA. As used herein, an “anterior direction” is in thedirection generally toward the cornea 14, while a “posterior direction”is generally in the direction toward the retina 12.

The natural lens 16 is contained within a capsular bag 20, which is athin membrane that completely encloses the natural lens 16 and isattached to a ciliary muscle 22 via zonules 24. An iris 26, disposedbetween the cornea 14 and the natural lens 16, provides a variable pupilthat dilates under lower lighting conditions (scotopic vision) andconstructs under brighter lighting conditions (photopic vision). Theciliary muscle 24, via the zonules 24, controls the shape and positionof the natural lens 16, which allows the eye 10 to focus on both distantand near objects. Distant vision is provided when the ciliary muscle 22is relaxed, wherein the zonules 24 pull the natural lens 16 so that thecapsular bag 20 is generally flatter and has a longer focal length(lower optical power). Near vision is provided as the ciliary musclecontracts, thereby relaxing the zonules 24 and allowing the natural lens16 to return to a more rounded, unstressed state that produces a shorterfocal length (higher optical power).

The optical performance of the eye 10 also depends on the spacingbetween the cornea 14 and the natural lens 16, sometimes referred to asthe anterior chamber depth prior to an ocular surgical procedure,ACD_(pre). As used herein, the “anterior chamber depth prior tosurgery”, “anterior chamber depth prior to an ocular surgicalprocedure”, or “ACD_(pre)”, is defined as a distance between an apex ofa cornea and an apex of a natural crystalline lens of an eye, prior to asurgery to replace the natural crystalline lens 16 with an intraocularlens. In some situations or cases, ACD_(pre) may be defined orapproximated as a distance between an apex of a cornea and an anteriorsurface of the iris 26.

Referring additionally to FIG. 2, which is a cross-sectional view of apseudophakic eye 10, the natural crystalline 16 lens has been replacedby an intraocular lens 100 according to an embodiment of the presentinvention. The intraocular lens 100 comprises an optic 102 and haptics104, the haptics 104 being generally configured to center the optic 102within the capsular bag 20. Numerous configurations of haptics 104relative to optic 102 are well know within the art and embodiments ofthe present invention may be applied to any of these.

In order to calculate, determine, or estimate the power of anintraocular lens 100 that is able to provide emmetropia or some otherpredetermined refractive outcome, various dimensions or measurements ofthe eye 10 are made prior to the surgical procedure. In addition toACD_(pre), embodiments of the present invention also measure axiallength AL of the eye 10, curvature of the cornea CR, is illustrated inFIG. 1.

Various formulations exist within the art that are used for calculationboth of lens power and position of an intraocular lens after an ocularsurgical procedure. These formulations generally comprise three steps:

1. Measure an eye.

2. Estimate the postoperative position of an intraocular lens.

3. Perform a lens power calculation based on the estimate and/or eyemeasurements.

The inventors have found that although all three steps are important,the second step of estimating the postoperative position of anintraocular lens may benefit most from improvements in the current stateof the measurement arts. For example, in the Norrby reference previouslycited in the Background section above, preoperative estimation ofpostoperative intraocular lens position was determined to be the largestcontributor of error in the refractive outcome of cataract surgery, withan error contribution of 35%, relative to all error sources evaluated.

Furthermore, the inventors have found that the combined measurements ofAL, ACD_(pre), and CR are highly predictive in calculating the positionof the implanted intraocular lens 100 or optic 102. As used herein, a“calculated position” or similar phrases, as applied to an IOL orophthalmic lens after implantation after an ocular surgical procedure,means an estimated or calculated distance between a cornea of an eye andthe IOL or lens after insertion, injection, or formation within the eye.The calculated position will generally be given herein in terms of the“postoperative anterior chamber depth” (ACD_(post)), which is definedherein as the distance from an apex of a cornea to an apex of animplanted intraocular lens. The calculated position may also be in termsof an “estimated lens position” (ELP), which is defined as a distancefrom an apex of a cornea to some effective lens plane of an implantedintraocular lens. For example, J. T. Holladay, M.D. defines ELP as adistance between an apex of a cornea and an effective principal plane ofan implantable intraocular lens (e.g., J. T. Holladay, M.D., et al.; JCataract Refract Surg; January 1988, Vol. 14, pp. 17-24, which is hereinincorporated by reference in its entirety). ELP is generally, though notnecessarily, greater than ACD_(post).

In certain embodiments, a highly predictive formulation of ACD_(post) iscalculated based on the mathematical relationship:

ACD_(post) =C1+(C2×AL)+(C3×ACD_(pre))+(C4×CR)  (1)

where AL is an the axial length of the eye, ACD_(pre) is the anteriorchamber depth prior to an ocular surgical procedure, CR is a radius ofcurvature of the cornea, and C1-C4 are constants.

In some embodiments, the calculated position of an IOL or ophthalmiclens after an ocular surgical procedure may be an estimated orcalculated distance between the cornea of an eye and a plane passingthrough the haptics of an ophthalmic lens, referred to herein and a lenshaptic plane. One of the co-inventors, Sverker Norrby, has developed athick-lens calculation scheme based on the lens haptic plane concept.See, for example, Norrby S.; J. Cataract Refractive Surgery 2004;30:1000-1005; Norrby S, Lydahl E, Koranyi G, J. Cataract RefractiveSurgery 2005; 31:1338-1344, U.S. Pat. No. 5,968,095; and U.S. PatentApplication Number 2007-0260157, all of which are herein incorporated byreference in their entirety. The lens haptic plane is defined herein asa plane where the IOL haptic makes contact with the eye tissue. As seenin FIG. 2, the lens haptic plane generally coincides with the equator ofthe capsular bag 20. As used herein, the distance between the cornea andthe lens haptic plane will be designated by “LHP”. The LHP may be anadditional independent parameter used in calculating or selecting a lenspower. It has been observed that the LHP is, in principle, generallyindependent of the power of an intraocular lens, while ACD_(post)generally depends more on power, although this generally produces only asmall shift. The LHP is the plane of contact with tissue (the equator ofthe capsular bag). Due to angulation of the haptic the anterior surfaceof an intraocular lens is generally posterior to LHP. The higher thepower, the thicker the intraocular lens. Therefore, the higher the lenspower, the more anterior is the anterior surface. Thus, the offsetbetween LHP and anterior surface depends on power.

In certain embodiments, a highly predictive formulation for a calculateddistance between the cornea and a lens haptic plane of a capsular bag ofan eye or of a haptic of an ophthalmic lens has the mathematicalrelationship:

LHP=C1′+(C2×AL)+(C3′×ACD_(pre))+(C4′×CR),  (2)

where AL is the axial length of the eye, ACD_(pre) is the anteriorchamber depth prior to an ocular surgical procedure, CR is a radius ofcurvature of the cornea, and C1′-C4′ are constants.

In one embodiment, the coefficients C1-C4 of Equation 1 were calculatedbased on a linear regression of data presented in Table 1 below. In thisembodiment, C1 has a nominal value of 4.236, C2 has a nominal value of0.08495, C3 has a nominal value of 0.42435, and C4 has a nominal valueof −0.3945. Thus, ACD_(post) is given by the equation:

ACD_(post)=4.236+(0.08495·AL)+(0.42435·ACD_(pre))−(0.3945·CR)  (3)

In other embodiments, each of the constants C1-C4 have values that arewithin plus or minus 20 percent of the nominal value shown in Equation3, within plus or minus 10 percent of the nominal value, or within plusor minus 5 percent of the nominal value.

In one embodiment, the coefficients C1′-C4′ were calculated based on alinear regression of data presented in Table 1 below. In thisembodiment, C1′ has a nominal value of 4.556, C2′ has a nominal value of0.08495, C3′ has a nominal value of 0.42435, and C4′ has a nominal valueof −0.3945. Thus, LHP is given by the equation:

LHP=4.556+(0.08495·AL)+(0.42435·ACD_(pre))−(0.3945·CR)  (4)

Referring to FIG. 3, in certain embodiments, a method 200 for selectingthe intraocular lens 100 or an optical power thereof comprises anelement 205 of determining one or more physical and/or opticalproperties of the eye 100. The method 200 also comprises an element 210of calculating a position of the intraocular lens 100 or the optic 102after an ocular surgical procedure. The method 200 additionallycomprises an element 215 of calculating or estimating an optical powerof the intraocular lens 100 suitable for providing a predeterminedrefractive outcome.

In the illustrated embodiment shown in FIGS. 1 and 2, element 205comprises measuring AL, ACD_(pre), and CR of the eye 10. Additionally oralternatively, a corneal power K may be measured or calculated based onCR, K may be expressed in units of Diopters (m⁻¹). In such embodiments,a conversion between CR and K is given by the relationship:

$\begin{matrix}{{K = {\frac{{KI} - 1}{CR} \times 1000}},} & (5)\end{matrix}$

where KI, the keratometric index, generally 1.3375. CR and K thuscontain the same information. In some embodiments, additional dimensionsof the eye may be measured (e.g., natural lens thickness LT, cornealthickness, retinal thickness, or the like). In addition, variousphysical property of the eye may also be measured or estimated (e.g., arefractive index of a material of the eye, and the like) and/or categoryinformation of the patient or IOL (e.g., age, sex, which eye, IOL model,IOL optic and/or haptic dimensions, or the like).

The element 210 of the method 200 comprises calculating a position ofthe intraocular lens 100 or the optic 102 after an ocular surgicalprocedure. In the illustrated embodiment of FIGS. 1 and 2, thecalculated position of the intraocular lens 100 is based on measured orcalculated values of AL, ACD_(pre), and CR of the eye 10. These valuesmay be used in one or more of Equations 1-4 to calculate the lensposition. In certain embodiments, the constants C1-C4 or C1′-C4′ areselected based on analysis of data for AL, ACD_(pre), and CR usingregression routine, for example, based on a multiple linear regression(MLR) analysis or a partial least squares (PLS) regression analysis ofdata for AL, ACD_(pre), and CR.

In the illustrated embodiment of FIGS. 1 and 2, Equations 1-4 have beenfound to be highly predictive on ACD_(post) and LHP, for example, havinga statistical P-Value of P>0.05, based on a PLS regression analysis ofdata from Table 1 below. In particular, Equation 2 has been found tohave a statistical P-Value of P less than or equal to 0.031. Based onthe PLS analysis for this data, Equation 3 has a relatively lowstatistical variation compared to other prior art formulations incalculating ACD_(post), for example, having Q2 variation of less than orequal to about 0.20, where “about” means±0.01, and a residual standarddeviation (RSD) of less than or equal to about 0.3, where “about”means±0.01. It is estimated that these results can improve differencebetween planned and final outcome, sometimes called mean absolute error(MAE), by 0.07 Diopters to a value of 0.053, based on data from theEuropean cataract outcome study (http://www.eurocat.net, accessed2008-07-10).

A characteristic of the mathematical relationship described by Equation2 is that the calculated value of ACD_(post) has a surprisingly lowsensitivity to variations in the measured axial length AL. As discussedbelow in the examples, the sensitivity of Equation 2 is only about 0.45%change in the calculated value of ACD_(post) when the value of ALchanges by 1% from a nominal value, for example a nominal value of 24 mmor 25 mm. This is much lower than the sensitivity of other evaluatedmathematical relationships based on different prior art combinations ofmeasured eye parameters, which varied from about 0.7% to about 2.4% whenthe value of AL is changed by 1% from a nominal value. Advantageously,the low dependence on AL of Equations 1-4 allows accurate calculationsof ACD_(post) to be made with less accurate measurements of AL.

Another surprising result from the study discussed in the Examples belowis that the addition of lens thickness LT to Equations 1-4 showedessentially no correlation of ACD_(post) with LT. This is in contrast toother prior art formulas (e.g, Olsen and Holladay) which include LTalong with AL, ACD_(pre), and CR in calculating ACD_(post). Thus, theformulas in Equations 1-4 advantageously determine ACD_(post) and LHPwithout the need to measure LT at all.

In some embodiments, one or more of the measured variables in Equations1-4 may be left out; however, this has been found to decrease thepredictability. For example, as discussed below in the Examples section,the measurement of CR may be left out and the coefficients for AL andACD_(pre) be re-evaluated. This can be useful for subjects that havealready had a corneal refractive procedure such as LASIK and there is norecord of the corneal curvature before the procedure.

The element 210 of the method 200 comprises element 210 of calculating aposition of the intraocular lens 100 or the optic 102 after an ocularsurgical procedure. Various power calculation formulas are available forthis purpose (e.g., Holladay 1, SRK/T, Hoffer Q, and the like).Generally, the lens power calculation only requires preoperativemeasurements are axial length AL and corneal power K (or equivalently,corneal curvature, CR), and are based on a thin-lens theoryapproximation for the optical calculation. The following representationis used by Holladay:

$P_{IOL} = {\frac{1336}{{AL} - {ELP}} - \frac{1336}{\frac{1336}{\frac{1000}{\frac{1000}{Rfx} - V} + K} - {ELP}}}$

where AL is the axial length in mm, K is the corneal power in Diopter ormm⁻¹, ELP is the estimated postoperative effective lens position in mm,V is the spectacle vertex distance in mm, and IOL is the IOL power inDiopters to obtain Rfx, the desired postoperative spectacle refraction.V is generally not critical and often assumed to be 12 mm.

In the case of emmetropia, the above equation may be reduced to:

$P_{IOL} = {\frac{1336}{{AL} - {ELP}} - \frac{1336}{\frac{1336}{K} - {ELP}}}$

In some embodiments, ACD_(post) may be substituted for ELP in the twoequations above.

The method 300 may be incorporated with one or more methods ofinserting, injecting, or forming a lens within the individual eyes ofthe population. Such methods may also comprise making postoperativemeasurements of the eyes in the population to determine thepostoperative position of the lens for each eye within the populationand/or to use the information to further refine the mathematical modesdefined by Equations 1-4. Additionally or alternatively, such methodsmay further comprise conducting a statistical analysis of each measuredor derived characteristic to determine (1) a correlation between thecalculated postoperative lens position and the measured or derivedcharacteristic(s) and/or (2) to determine coefficient value for anequation containing the measured or derived characteristic(s) asvariables, the equation configured for calculating a postoperative lensposition within an eye.

In certain embodiments, a method of selecting an ophthalmic lens or anoptical power thereof comprises determining an axial length and/or othercharacteristic of an eye having a cornea. The method also comprisescalculating a position of an ophthalmic lens after an ocular surgicalprocedure and calculating an optical power of the ophthalmic lenssuitable for providing a predetermined refractive outcome. Thecalculated position of the ophthalmic lens is based on a mathematicalrelationship that includes an input parameter representing the axiallength of the eye, the mathematical relationship having a sensitivitythat is less than or equal to 0.4 percent.

Referring to FIG. 4, in certain embodiments, a computer system 300 forcalculating a postoperative lens position within an eye and/or forselecting an ophthalmic lens or an optical power thereof comprises aprocessor 302 and a computer readable memory 304 coupled to theprocessor 302. The computer readable memory 304 has stored therein anarray of ordered values 308 and sequences of instructions 310 which,when executed by the processor 302, cause the processor 302 to calculatea postoperative lens position within an eye and/or for selecting anophthalmic lens or an optical power thereof. The array of ordered values308 may comprise, for example, one or more ocular dimensions of an eyeor plurality of eyes from a database, a desired refractive outcome,parameters of an eye model based on one or more characteristics of atleast one eye, and data related to an IOL or set of IOLs such as apower, an aspheric profile, and/or a lens plane. In some embodiments,the sequence of instructions 310 includes determining a location of thelens plane of an IOL, performing one or more calculations to determine apredicted refractive outcome based on an eye model and a ray tracingalgorithm, comparing a predicted refractive outcome to a desiredrefractive outcome, and based on the comparison, repeating thecalculation with an IOL having at least one of a different power, adifferent aspheric profile, and a different lens plane.

The computer system 300 may be a general purpose desktop or laptopcomputer or may comprise hardware specifically configured performing thedesired calculations. In some embodiments, the computer system 300 isconfigured to be electronically coupled to another device such as aphacoemulsification console or one or more instruments for obtainingmeasurements of an eye or a plurality of eyes. In other embodiments, thecomputer system 300 is a handheld device that may be adapted to beelectronically coupled to one of the devices just listed. In yet otherembodiments, the computer system 300 is, or is part of, refractiveplanner configured to provide one or more suitable intraocular lensesfor implantation based on physical, structural, and/or geometriccharacteristics of an eye, and based on other characteristics of apatient or patient history, such as the age of a patient, medicalhistory, history of ocular procedures, life preferences, and the like.

Generally, the instructions of the system 300 will include elements ofthe method 300 and/or parameters and routines for performingcalculations of one or more of Equations 1-4.

In certain embodiments, the system 300 includes or is part aphacoemulsification system, laser treatment system, optical diagnosticinstrument (e.g, autorefractor, aberrometer, and/or corneal topographer,or the like). For example, the computer readable memory 304 mayadditionally contain instructions for controlling the handpiece of aphacoemulsification system or similar surgical system. Additionally oralternatively, the computer readable memory 304 may additionally containinstructions for controlling or exchanging data with an autorefractor,aberrometer, and/or corneal topographer, or the like,

In some embodiments, the system 300 includes or is part of a refractiveplanner. The refractive planner may be a system for determining one ormore treatment options for a subject based on such parameters as patientage, family history, vision preferences (e.g., near, intermediate,distant vision), activity type/level, past surgical procedures.

EXAMPLES

Various formulas exist in the art for calculating intraocular lens powerfor an individual subject or patient. In general, these formulationsinclude equations for estimating ELP. Many of these formulas are basedon thin-lens theory, in which each lens is reduced to a power at itsprincipal plane. The position of the principal plane may depend on theshape factor of the lens or on the vergence of the incoming light, henceELP is not generally indicative of the true position of the intraocularlens. The formulas differ from each other in the algorithms forestimation of ELP. These algorithms were obtained by analysis of largeamounts of clinical data and adjusted such that the calculated refractedoutcome on average agreed with that found postoperatively.

To compensate for deviations in unusually long or unusually short eyes,many formulas also include certain “fudge” factors. For example, theHolladay 1, SRK/T, Hoffer Q formulas include a SF-, A- and ACD-constant,respectively, that is specific for a given intraocular lens model and isassociated with an average postoperative position of that model.Manufactures may give preliminary estimates of these constants, whichmay then be “personalized” by or for a particular surgeon. To“personalize” a constant for a given formula the surgeon may performseveral cases with the preliminary constant, and subsequently adjuststhe constant such that the mean difference with refractive outcomesbecomes zero. Derivation of these formula algorithms is generally basedon back-calculation of clinical outcome using thin-lens theory. Theymay, therefore, be confounded with all errors of the measured parametersinvolved, and the oversimplification of thin-lens theory.

The Haigis formula is based on thin-lens theory, but has a somewhatdifferent approach. It uses three constants in the algorithm for ELP,two of which are multiplicative coefficients for AL and ACD. By usingthree constants it can better approximate the nonlinear relationshipbetween AL and ELP. Olsen has presented a formula based on thick-lenstheory, though still assuming the paraxial approximation. It uses 5parameters to predict the postoperative position of the anterior surfaceof the IOL, ACD_(post) (in mm): AL, ACD, lens thickness (LT in mm),corneal radius (CR in mm), and corneal diameter (“white-to-white” inmm). The advantage with the thick-lens model is that the opticalcomponents are represented by the refracting surfaces at their realpositions. The challenge is that all input parameters must be correct.There are no “fudge” factors to tweak the results with, except theACD_(post) prediction algorithm itself. The Holladay 2 formula, whichhas not been published, resides in the Holladay IOL Consultant softwareand can use up to 7 parameters in the ELP prediction algorithm: AL, K,ACD, LT, corneal diameter, preoperative refraction in Diopters, andpatient age (in years). It is still apparently based on thin-lenstheory.

In an effort to evaluate various types of formulas for calculating theposition of an implanted intraocular lens, a study was conducted, inwhich 23 regular cataract patients received either an AMO CeeOn 911A oran AMO TECNIS Z9000 IOL. These lenses have different shapes of theanterior optic surface, but are otherwise identical, and thereforeprovide the same axial depth in a given eye. Patients were treated inaccordance with normal hospital routines; however, additionalpreoperative and postoperative measurements were performed for thepurposes of the study. Zeiss IOLMaster, Zeiss ACMaster (bothmanufactured by Carl Zeiss Meditec, Inc., Dublin, Calif.). Measurmentsincluded axial length, anterior chamber depth, lens thickness and meancorneal radius. Postoperative anterior chamber depth was measured withZeiss ACMaster. Formulas to estimate postoperative anterior chamberdepth in terms of preoperative parameters were analyzed by means ofpartial least squares regression. The usefulness of fits was judged bythe P-value obtained by cross-validation ANOVA.

TABLE 1 Measured eye parameters, implanted intraocular lens, andpostoperative refraction for study used to determine. Post-surg.Categories Pre-surgery measurement Measurement Age AL.i ACD.a ACD.i CR.iLT.a WTW.a ACD_(post).a Case Sex Eye Model (yrs) (mm) (mm) (mm) (mm)(mm) (mm) (mm) 1 m Left C 67 24.1 3.76 3.69 8.1 4.3 12.6 4.61 2 f Left C78 23.52 3.56 3.52 7.8 4.64 12.9 4.65 4 m Right C 62 24.17 3.61 3.6 7.964.02 11.7 4.60 6 f Right C 70 24.73 3.48 3.59 7.79 4.23 11.8 4.92 7 fLeft C 75 22.98 3.64 3.43 7.64 4.11 12.8 4.39 8 f Right C 66 23.64 3.083.04 7.85 4.24 10.9 4.28 9 m Right C 76 23.07 3.66 3.54 7.78 4.65 10.94.73 10 f Left C 76 23.13 3.7 3.23 7.67 4.51 11.2 4.30 11 f Right C 7523.29 3.41 3.79 7.53 4.98 11.7 4.83 12 f Right T 67 22.13 3.28 3.33 7.34.49 12.8 4.48 13 f Left T 77 22.34 3.75 3.23 7.49 4 12.7 4.60 14 f LeftT 67 24.32 3.26 3.21 7.95 4.68 12.9 4.29 15 f Left T 77 22.13 2.77 2.597.48 4.82 12.6 4.61 16 f Left T 74 24.83 2.96 2.8 7.44 4.99 12 4.60 17 mLeft T 65 23.39 3.66 3.22 7.44 4.55 11.3 4.80 18 f Right T 64 23.2 3.082.79 7.79 4.98 12.4 4.24 19 f Left T 74 23.28 3.33 3.16 7.48 4.15 12.14.94 20 m Left T 44 24.53 3.63 3.38 8.3 3.7 14 4.21 21 m Left T 76 24.713.66 3.56 8.14 4.56 11.8 4.93 22 f Left T 69 22.31 2.91 2.85 7.99 4.2512.5 3.98 23 f Left T 79 22.53 2.65 2.59 7.53 4.98 13.3 4.53 24 f RightT 39 22.03 3.45 3.19 7.61 4.07 12.6 4.58 Mean: 69 23.21 3.38 3.24 7.734.45 12.3 4.52 SD: 10 0.91 0.34 0.35 0.27 0.37 0.8 0.26 In the “Model”column, C = AMO CeeOn 911A intraocular lens and T = AMO TECNIS Z9000intraocular lens. Under “Pre-surgery” and “Post-surg.”, “.i” and “.o”indicate measurements made using an ACMaster and IOLMaster,respectively.

Of the 23 patients enrolled in the study, 10 received a CeeOn 911A IOLand 13 a Tecnis Z9000 IOL. One patient in the former group did notcomplete the study, leaving a total of 22 cases for analysis. Data ofthis type are frequently analyzed by multiple linear regression (MLR);however, such analysis may be valid only if the parameters are notcorrelated to each other. When the parameters are correlated, partialleast squares (PLS) regression is considered a better approach. PLSregression is further capable of handling category variables, such assex, and can cope with missing data. Accordingly, data in this study wasanalyzed by PLS regression using SIMCA P 12+ (Umetrics AB, Umeå,Sweden). In accordance with the manual of SIMCA, each combination ofinput parameters is referred to as a “model”, not to be confused withIOL model.

For small data sets, as in this study, the prediction power of a modelwas tested by cross validation by elimination of one observation at atime; thereafter a new model was calculated. This new sub-model was usedto predict the observation eliminated. The process was continued untilall observations were eliminated once. An average prediction error wasthen calculated from the sum of squares of all prediction errors of thesub-models. This was used to calculate the statistic Q², which is anestimate of how much of the variation in the result parameter(ACD_(post) in this case) can be predicted by the model parameters at achosen level of certainty. A 95% level was chosen for this study. Thistype of cross validation (CV), also known as “Jack-Knifing”, deliversuncertainty intervals for each model variable. CV-ANOVA of theprediction errors yields the residual standard deviation (RSD), and thereliability of the PLS-models as a P-value. As commonly done we considerP<0.05 as statistically significant.

Of the various models evaluated, some of the more relevant ones arereported in Table 2. Some of models shown in Table 2 represent some ofthe more common measurement combination used in estimating ACD_(post).However, of all the models shown, only model number 8 (which correspondsto embodiments of the present invention discussed in relation toEquations 1-4) proved to be clearly statistically significant (e.g.,having a value of P>0.05). This model also had a reasonably low RSD of0.20.

TABLE 2 Statistical results for various combinations of pre-surgicallymeasured eye dimensions. Model RSD No. Preoperative input variables Q2 P(mm) 4 Age, AL.i, ACD.a, CR.i, LT.a, WTW.a 0.055 0.582 0.22 5 Age, AL.i,ACD.a, CR.i, WTW.a 0.061 0.549 0.21 6 AL.i, ACD.a, CR.i, WTW.a 0.0870.422 0.22 7 AL.i, ACD.a, CR.i 0.230 0.083 0.21 8 AL.i, ACD.i, CR.i0.305 0.031 0.20 10 AL.i, ACD.i, CR.i, AL.i · ACD.i, ACD.i · CR.i 0.4170.182 0.15 12 AL.i, ACD.i 0.127 0.275 0.24 14 AL.i, CR.i 0.011 1.0000.23 16 AL.i, ACD.i, CR.i, AL.i · ACD.i 0.467 0.083 0.16 17 AL.i, K.i0.011 1.000 0.23 26 AL.i, ACD.a, CR.i, LT.a 0.228 0.086 0.20 “.i” and“.o” indicate measurements made using an ACMaster and IOLMaster,respectively.

For the M8 model, there was a generally strong correlation withACD_(pre) and CR. Surprisingly, however, there was only a weakcorrelation with axial length AL. The generally low correlation to AL issurprising, since AL is generally considered as a strong predictor forpostoperative intraocular lens position. In present formulas thealgorithms for prediction of the postoperative IOL position, or theACD_(post) were obtained by back-calculation by means of the opticalformula. The strong influence AL has on the power calculation itself,probably explains why it then becomes a strong predictor of IOLposition.

A sensitivity analysis was also performed for the various terms used inEquations 3 and 4 above, for those found in the currently popularformations discussed above, and for a formula used in U.S. PatentApplication Publication Number 2007/0260157 (e.g., the equation shown inparagraph [0061]). To that end, values of the ACD_(post) estimate foreach formulation calculated for an average eye model with the followingcharacterization:

AL=24.75 mm

ACD_(pre)=3.54 mm

CR=8 mm (or K=42.2 Diopters)

Based on these nominal values, an ACD_(post) calculation was made foreach of the formulations cited below in Table 3. Additional ACD_(post)calculations were also made for each formulation in which each of theparameters for each formula was increased by 1 percent above the nominalvalues above. In this way, a percent change in the calculated ACD_(post)was found for a 1 percent increase in the different parameters used ineach formulation. The resulting percent change in the ACD_(post)estimate for the various 1 percent parameter changes is shown in Table3.

TABLE 3 Calculated sensitivity of the parameters in various formulas forused to estimate ACD_(post). AL ACD.pre CR Rostock (Eq. (3)) 0.45 0.32−0.67 2007/0260157 publication 1.20 −0.75 −0.75 Holladay 2.42 — −1.49Haigis 0.73 0.15 — SRK/T 0.96 — −1.47 Hoffer 1.25 — −0.40

The formulation for the 2007/0260157 publication is:

LHP=2.486+[0.2174×(AL+ΔAL)]−(0.4213×CR),

where ΔAL is a retinal thickness parameter and is equal to 0.25 mm.

As can be seen from the results in Table 3, the Rostock equation variesonly 0.45 percent when the axial length AL parameter is increased by 1percent from the nominal value. This confirms that the Rostockformulation provides relationships that has a low sensitivity to thevalue of the axial length AL of the eye. By contrast, the otherformulations that were analyzed showed a much higher sensitivity to thevalue of axial length AL. The least sensitive of these remainingformulations was that of Haigis, which was still more than three timesas sensitive as the Rostock equation, Equation 3. These resultsdemonstrate that the Rostock formulation clearly has a very lowsensitivity to the value of axial length AL of an eye.

Prediction of postoperative intraocular lens position in cases where asubject has had a previous corneal refractive surgery is an everincreasing problem. The final postoperative position of the intraocularlens is not generally affected by the fact that the patient has hadprevious corneal surgery. But CR and K have been altered and, therefore,formulas that use either CR or K are no longer valid for these patientswhere the preoperative values have been lost. In such cases, the Rostockformulation can be adjusted to produce modified Rostock formulas thatare independent of CR:

ELP=3.18+(0.0203×AL)+(0.281×ACD_(pre))  (6)

LHP=3.50+(0.0203×AL)+(0.281×ACD_(pre))  (7)

The above presents a description of the best mode contemplated ofcarrying out the present invention, and of the manner and process ofmaking and using it, in such full, clear, concise, and exact terms as toenable any person skilled in the art to which it pertains to make anduse this invention. This invention is, however, susceptible tomodifications and alternate constructions from that discussed abovewhich are fully equivalent. Consequently, it is not the intention tolimit this invention to the particular embodiments disclosed. On thecontrary, the intention is to cover modifications and alternateconstructions coming within the spirit and scope of the invention asgenerally expressed by the following claims, which particularly pointout and distinctly claim the subject matter of the invention.

1. A method of selecting an intraocular lens or an optical powerthereof, comprising: providing an eye comprising a cornea and a retina;determining a value of an axial length of an eye, the axial length beingequal to a distance from the apex to a surface of the retina;calculating, based on a mathematical relationship, a distance from theapex to a plane the intraocular lens after an ocular surgical procedure;calculating an optical power of the intraocular lens suitable forproviding a predetermined refractive outcome; wherein the mathematicalrelationship comprises a dependent variable that varies with a value ofan independent variable, the independent variable having a base valuecorresponding to the determined axial length; wherein the mathematicalrelationship has a sensitivity of less than or equal to 0.5%, thesensitivity being defined as a percent variation in the dependentvariable for a 1% in the base value.
 2. The method of claim 1, whereinthe mathematical relationship has a sensitivity of less than or equal to0.6%
 3. The method of claim 1, further comprising at least one ofplacing the intraocular lens behind the cornea, within the cornea, or ontop of the cornea, or forming the intraocular lens from an injectablematerial behind the cornea, within the cornea, or on top of the cornea.4. The method of claim 1, wherein the sensitivity is less than or equalto 0.4 percent when the method is used to calculate a position of TecnisZ9000 intraocular lens after an ocular surgical procedure.
 5. The methodof claim 1, wherein the mathematical relationship includes an anteriorchamber depth prior to an ocular surgical procedure.
 6. The method ofclaim 5, wherein the distance from the apex to a plane the intraocularlens after an ocular surgical procedure is a dependent variableACD_(post) and the mathematical relationship is:ACD_(post) =C1+C2×AL+C3×ACD_(pre) +C4×CR, where AL is an input parameterrepresenting the axial length of the eye, ACD_(pre) is an inputparameter representing the anterior chamber depth prior to an ocularsurgical procedure, CR is an input parameter representing a radius ofcurvature of the cornea, C1 is a constant with a nominal value of 4.236,C2 is a constant with a nominal value of 0.08495, C3 is a constant witha nominal value of 0.42435, and C4 is a constant with a nominal value of−0.3945, each of the constants having a value that is within plus orminus 20 percent of the nominal value thereof.
 7. The method of claim 5,wherein each of the constants has a value that is within plus or minus 5percent of the nominal value thereof.
 8. The method of claim 5, whereinC1 equals 4.236, C2 equals 0.08495, C3 equals 0.42435, and C4 equals−0.3945.
 9. The method of claim 5, wherein the calculated position is acalculated distance between the cornea and a lens haptic plane of anintraocular lens and the mathematical relationship is:LHP=C1+C2×AL+C3×ACD_(pre) +C4×CR, where LHP is the calculated distancebetween the cornea and a lens haptic plane, AL is an input parameterrepresenting the axial length of the eye, ACD_(pre) is an inputparameter representing the anterior chamber depth prior to an ocularsurgical procedure, CR is an input parameter representing a radius ofcurvature of the cornea, C1 is a constant with a nominal value of 4.556,C2 is a constant with a nominal value of 0.08495, C3 is a constant witha nominal value of 0.42435, and C4 is a constant with a nominal value of−0.3945, each of the constants having a value that is within plus orminus 20 percent of the nominal value thereof.
 10. The method of claim5, wherein the calculated position is a calculated anterior chamberdepth, ACD_(post), and the mathematical relationship is:ACD_(post) =C1+C2×AL+C3×ACD_(pre), where AL is an input parameterrepresenting the axial length of the eye, ACD_(pre) is an inputparameter representing the anterior chamber depth prior to an ocularsurgical procedure, C1 is a constant with a nominal value of 3.18, C2 isa constant with a nominal value of 0.0203, and C3 is a constant with anominal value of 0.281, each of the constants having a value that iswithin plus or minus 20 percent of the nominal value thereof.
 11. Themethod of claim 5, wherein the calculated position is a calculateddistance between the cornea and a lens haptic plane of an intraocularlens and the mathematical relationship is:LHP=C1+C2×AL+C3×ACD_(pre), where LHP is the calculated distance betweenthe cornea and a lens haptic plane, AL is an input parameterrepresenting the axial length of the eye, ACD_(pre) is an inputparameter representing the anterior chamber depth prior to an ocularsurgical procedure, C1 is a constant with a nominal value of 3.50, C2 isa constant with a nominal value of 0.0203, and C3 is a constant with anominal value of 0.281, each of the constants having a value that iswithin plus or minus 20 percent of the nominal value thereof.
 12. Themethod of claim 1, wherein the mathematical relationship includes aradius of curvature of the cornea.
 13. The method of claim 1, whereinthe mathematical relationship is independent of a radius of curvature ofthe cornea.
 14. The method of claim 13, wherein calculating an opticalpower of the intraocular lens comprises calculating an optical power ofan intraocular lens for use with a subject that has previously undergonea corneal refractive procedure.
 15. The method of claim 1, wherein themathematical relationship includes measured or calculated parameters ofthe eye that are multiplied together.
 16. A system for providing anintraocular lens, the system comprising: a processor; and a computerreadable memory configured to communicate with the processor, the memoryhaving stored therein: an array of ordered values, including at leastone of: one or more ocular dimensions, the ocular dimensions includingan axial length of an eye having a cornea with an apex and a retinasurface, the axial length being equal to a first distance from the apexto the retina surface; a predetermined refractive outcome; a sequence ofinstructions which, when executed by the processor, cause the processorto select an intraocular lens, select an optical power of theintraocular lens, or provide the intraocular lens, the sequence ofinstructions including: calculating, based on a mathematicalrelationship, a second distance from the apex to an apex or a plane ofthe intraocular lens when the intraocular lens is inserted into the eye;calculating an optical power of the intraocular lens suitable forproviding the predetermined refractive outcome; wherein the mathematicalrelationship comprises a dependent variable that varies with a value ofan independent variable, the independent variable having a base valuecorresponding to the axial length of the eye; wherein the mathematicalrelationship has a sensitivity of less than or equal to 0.5%, thesensitivity defined as a percent variation in the dependent variable fora 1% increase of the independent variable from the base value.
 17. Asystem for providing an intraocular lens, the system comprising: aprocessor; and a computer readable memory configured to communicate withthe processor, the memory having stored therein: an array of orderedvalues, including at least one of: one or more ocular dimensions of aneye having a cornea with an apex and a retina surface; a predeterminedrefractive outcome; a sequence of instructions which, when executed bythe processor, cause the processor to select an intraocular lens, selectan optical power of the intraocular lens, or provide the intraocularlens, the sequence of instructions including: calculating, based on amathematical relationship, a distance from the apex to an apex or planeof the intraocular lens when the intraocular lens is inserted into theeye; calculating an optical power of the intraocular lens suitable forproviding the predetermined refractive outcome; wherein the mathematicalrelationship includes an independent variable that depends on an axiallength of the eye, AL, an anterior chamber depth of the naturalcrystalline lens, ACD_(pre), and a corneal radius of the eye, CR, but isindependent of a thickness of the natural crystalline lens, LT.
 18. Thesystem of claim 17, wherein the sequence of instructions furthercomprises: determining a location of the plane of the intraocular lens;performing a calculation to determine a predicted refractive outcomebased on the eye model and a ray tracing algorithm; comparing thepredicted refractive outcome to the desired refractive outcome; based onthe comparison, repeating the performed calculation with an IOL havingat least one of a different power, a different aspheric profile, and adifferent lens plane.
 19. The system of claim 17, wherein the systemcomprises a phaco emulsification system.
 20. The system of claim 17,wherein the calculated distance is a calculated anterior chamber depthafter a surgery, ACD_(post), and the mathematical relationship is:ACD_(post) =C1+C2×AL+C3×ACD_(pre) +C4×CR, where AL is an input parameterrepresenting the axial length of the eye, ACD_(pre) is an inputparameter representing the anterior chamber depth prior to an ocularsurgical procedure, CR is an input parameter representing a radius ofcurvature of the cornea, and C1-C4 are constants.
 21. The system ofclaim 20, wherein C1 equals 4.236, C2 equals 0.08495, C3 equals 0.42435,and C4 equals −0.3945.
 22. The system of claim 21, wherein each of theconstants has a value that is within plus or minus 5 percent of thenominal value thereof.
 23. The system of claim 17, wherein thecalculated distance is a calculated distance between the cornea and alens haptic plane, LHP, of an intraocular lens and the mathematicalrelationship is:LHP=C1+C2×AL+C3×ACD_(pre) +C4×CR, where LHP is the calculated distancebetween the cornea and a lens haptic plane, AL is an input parameterrepresenting the axial length of the eye, ACD_(pre) is an inputparameter representing the anterior chamber depth prior to an ocularsurgical procedure, CR is an input parameter representing a radius ofcurvature of the cornea, C1 is a constant with a nominal value of 4.556,C2 is a constant with a nominal value of 0.08495, C3 is a constant witha nominal value of 0.42435, and C4 is a constant with a nominal value of−0.3945, each of the constants having a value that is within plus orminus 20 percent of the nominal value thereof.
 24. The system of claim23, wherein the calculated position is a calculated anterior chamberdepth, ACD_(post), and the mathematical relationship is:ACD_(post) =C1+C2×AL+C3×ACD_(pre), where AL is an input parameterrepresenting the axial length of the eye, ACD_(pre) is an inputparameter representing the anterior chamber depth prior to an ocularsurgical procedure, C1 is a constant with a nominal value of 3.18, C2 isa constant with a nominal value of 0.0203, and C3 is a constant with anominal value of 0.281, each of the constants having a value that iswithin plus or minus 20 percent of the nominal value thereof.
 25. Thesystem of claim 17, wherein the calculated position is a calculateddistance between the cornea and a lens haptic plane of an intraocularlens and the mathematical relationship is:LHP=C1+C2×AL+C3×ACD_(pre), where LHP is the calculated distance betweenthe cornea and a lens haptic plane, AL is an input parameterrepresenting the axial length of the eye, ACD_(pre) is an inputparameter representing the anterior chamber depth prior to an ocularsurgical procedure, C1 is a constant with a nominal value of 3.50, C2 isa constant with a nominal value of 0.0203, and C3 is a constant with anominal value of 0.281, each of the constants having a value that iswithin plus or minus 20 percent of the nominal value thereof.